Finite Group Databasebeta

← #44562388

Subgroups — #44562388

Order 32128.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632377C211yesmaximal, characteristic, complemented
2D4×D1004C211yesmaximal, characteristic
3#30632052C211yesmaximal, characteristic, complemented
4#30632012C211yesmaximal, characteristic, complemented
5#30632001C211yesmaximal, characteristic, complemented
6#30631988C211yesmaximal, characteristic, complemented
7#30631933C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631888C211yesmaximal, characteristic, complemented
10#30631840C211yesmaximal, characteristic, complemented
11#30631840C211yesmaximal, characteristic, complemented
12#30631789C211yesmaximal, characteristic, complemented
13#30631693C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×D4×D251—21no—
19C2×#16860799C2211yescharacteristic
20C2×#16860799C2211yescharacteristic, complemented
21C2×#16860799C2211yescharacteristic, complemented
22#22548712C2211yescharacteristic
23#22548708—21no—
24#22548707C2211yescharacteristic, complemented
25#22548705—21no—
26#22548696—21no—
27#22548694C2211yescharacteristic, complemented
28#22548694C2211yescharacteristic, complemented
29#22548690—21no—
30#22548689C2211yescharacteristic
31#22548689C2211yescharacteristic
32#22548685—21no—
33#22548685—21no—
34C4×Dic502C2211yescharacteristic, complemented
35C4×D1004C2211yescharacteristic
36#22548674C2211yescharacteristic, complemented
37#22548671—21no—
38C2×C4⋊Dic251C2211yescharacteristic, complemented
39#22548649C2211yescharacteristic, complemented
40#22548649C2211yescharacteristic, complemented
41#22548647C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic, complemented
47#22548637C2211yescharacteristic, complemented
48#22548637C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic, complemented
52C2×M16×C251C2211yescharacteristic, complemented
53C22×C8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic, complemented
56C22⋊C8×C251C2211yescharacteristic, complemented
57C22⋊C8×C251C2211yescharacteristic, complemented
58#97×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C23×D251—21no—
61C2×Dic502C2311yescharacteristic
62C2×D1004—22no—
63C2×D1004—22no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—21no—
67C2×D1004—21no—
68C2×D1004—41no—
69C2×D1004—41no—
70C2×D1004C2311yescharacteristic
71C2×D1004C2311yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75D4×D251—44no—
76C22×Dic251—21no—
77C2×C4×D251—21no—
78#16860799—22no—
79#16860799—41no—
80#16860799—41no—
81C4⋊Dic251—21no—
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97C4×Dic251—21no—
98C4×Dic251—21no—
99C2×D4×C251D411yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102M16×C251—41no—
103C2×C8×C251—41no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C22⋊C4×C251D411yescharacteristic
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—42no—
118C22×D251—42no—
119C22×D251—81no—
120C22×D251—81no—
121Dic502—21no—
122Dic502—21no—
123D1004—21no—
124D1004—21no—
125D1004—21no—
126D1004—21no—
127D1004—21no—
128D1004—42no—
129D1004—42no—
130D1004—42no—
131D1004—81no—
132D1004#4112yes—
133C22⋊D251—42no—
134C22⋊D251—42no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—42no—
141C4×D251—42no—
142C23×C251C2×D411yescharacteristic
143C23×C251C2×D411yescharacteristic
144D4×C251—22no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2×D411yescharacteristic
153C2×C4×C251C2411yescharacteristic
154C8×C251—21no—
155C8×C251—21no—
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—21no—
163C22×C251AGL(1,ℤ/8)12yes—
164C22×C251C22×D411yescharacteristic
165C2×D251—22no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—84no—
175C2×D251—84no—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—21no—
179C4×C251—21no—
180C4×C251C2×#4111yescharacteristic
181C4×C251H(5,𝔽2)11yescharacteristic
182Dic251—41no—
183Dic251—41no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187C2×C251—22no—
188C2×C251—41no—
189C2×C251#54611yescharacteristic
190C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C251#57711yescharacteristic
192D251—42no—
193D251—81no—
194D251—81no—
195C251#331011yescharacteristic, complemented, Sylow
196#3310—2511nomaximal, Sylow
197#552—2511no—
198D42—2511no—
199#490—2511no—
200#482—2511no—
201#480—2511no—
202#478—2511no—
203#468—2511no—
204#465—2511no—
205#460—2511no—
206#450—2511no—
207#450—2511no—
208#440—2511no—
209#435D25111yescharacteristic, complemented
210#418—2511no—
211C2×#102—2511no—
212C22×D4—5021no—
213C22×D4—2511no—
214C22×D4—2511no—
215C2×SD16—2511no—
216C2×SD16—2511no—
217C2×SD16—2511no—
218C2×M16C2×D25111yescharacteristic, complemented
219C22×C8C2×D25111yescharacteristic, complemented
220#127—2511no—
221#126—5021no—
222#125—5021no—
223#125—2511no—
224#123—5021no—
225C22⋊Q8—2511no—
226C22⋊Q8—2511no—
227#121—5021no—
228#121—2511no—
229#121—2511no—
230#120—5021no—
231#120—5021no—
232C4×Q8—2511no—
233C4×D4—2511no—
234C4×D4C2×D25111yescharacteristic
235#117—5021no—
236#117—2511no—
237C2×C4⋊C4—2511no—
238#107—2511no—
239#107—2511no—
240#106—2511no—
241C4⋊C8C2×D25111yescharacteristic, complemented
242#103—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102—2511no—
251#102—2511no—
252C22⋊C8C2×D25111yescharacteristic, complemented
253C22⋊C8C2×D25111yescharacteristic, complemented
254#97C2×D25111yescharacteristic, complemented
255C24—5021no—
256C24—5021no—
257C2×Q8—2511no—
258C2×D4—10041no—
259C2×D4—10041no—
260C2×D4—10044no—
261C2×D4—5022no—
262C2×D4—5022no—
263C2×D4—5021no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—5021no—
268C2×D4—5021no—
269C2×D4—2511no—
270C2×D4—2511no—
271C2×D4—2511no—
272C2×D4D100411yescharacteristic
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277SD16—10041no—
278SD16—10041no—
279SD16—5022no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D25111yescharacteristic
283C2×C8C22×D25111yescharacteristic
284C2×C8C22×D25111yescharacteristic
285C2×C8C22×D25111yescharacteristic
286C4⋊C4—5021no—
287C4⋊C4—5021no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—5021no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4—2511no—
295C4⋊C4—2511no—
296C4⋊C4D100411yescharacteristic
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4D100411yescharacteristic
304C22⋊C4D100411yescharacteristic
305C42—5021no—
306C42—5021no—
307C42C22×D25111yescharacteristic
308C23—20081no—
309C23—20081no—
310C23—10042no—
311C23—10042no—
312C23—10042no—
313C23—10042no—
314C23—5021no—
315C23—5021no—
316C23—5021no—
317C23C2×D100411yescharacteristic
318C23C2×D100411yescharacteristic
319Q8—5021no—
320Q8—5021no—
321D4—22no—
322D4—20081no—
323D4—10042no—
324D4—10042no—
325D4—10042no—
326D4—10042no—
327D4—10042no—
328D4—5021no—
329D4—5021no—
330D4—5021no—
331D4—5021no—
332D4—5021no—
333D4—2512no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—10042no—
339C2×C4—10042no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4C2×D100411yescharacteristic
346C2×C4C2×D100411yescharacteristic
347C2×C4C2×D100411yescharacteristic
348C2×C4C2×D100411yescharacteristic
349C2×C4C23×D25111yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—20084no—
360C22—20084no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22—10041no—
369C22—5022no—
370C22#2254874312yes—
371C22C22×D100411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—10041no—
377C4—10041no—
378C4—10041no—
379C4—10041no—
380C4—10041no—
381C4#41×D25111yescharacteristic
382C4#2254876011yescharacteristic
383C2—22no—
384C2—41no—
385C2—20081no—
386C2—20081no—
387C2—10042no—
388C2#3063233211yescharacteristic
389C2C2×#2254874311yescharacteristic
390C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#435×C251C2
socleC22×C251C22×D4
prime radicalD4×D1004C2
hypercenter#435D251
hypocenterC251#3310
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #30632377C2semidirect
3. #30632052C2semidirect
4. #30632012C2semidirect
5. #30632001C2semidirect
6. #30631988C2semidirect
7. #30631933C2semidirect
8. #30631916C2semidirect
9. #30631888C2semidirect
10. #30631840C2semidirect
11. #30631840C2semidirect
12. #30631789C2semidirect
13. #30631693C2semidirect
14. C2×#22548637C2semidirect
15. #435×C251C2semidirect
20. C2×#16860799C22semidirect
21. C2×#16860799C22semidirect
24. #22548707C22semidirect
27. #22548694C22semidirect
28. #22548694C22semidirect
34. C4×Dic502C22semidirect
36. #22548674C22semidirect
38. C2×C4⋊Dic251C22semidirect
39. #22548649C22semidirect
40. #22548649C22semidirect
41. #22548647C22semidirect
42. #22548640C22semidirect
46. #22548637C22semidirect
47. #22548637C22semidirect
48. #22548637C22semidirect
49. #22548637C22semidirect
50. #22548637C22semidirect
51. #22548637C22semidirect
52. C2×M16×C251C22semidirect
53. C22×C8×C251C22semidirect
55. C4⋊C8×C251C22semidirect
56. C22⋊C8×C251C22semidirect
57. C22⋊C8×C251C22semidirect
58. #97×C251C22semidirect
195. C251#3310semidirect
209. #435D251semidirect
218. C2×M16C2×D251semidirect
219. C22×C8C2×D251semidirect
241. C4⋊C8C2×D251semidirect
252. C22⋊C8C2×D251semidirect
253. C22⋊C8C2×D251semidirect
254. #97C2×D251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306323772
D4×D10042
#306320522
#306320122
#306320012
#306319882
#306319332
#306319162
#306318882
#306318402
#306317892
#306316932
C2×#225486372
#435×C2512
#3310251